On the Variational Problems without Having Desired Variational Symmetries

Abstract

We will have an attempt to present a method for constructing variational problems without having a desired one-parameter transformation group as a variational symmetry. For this, we use the notation of μ-symmetry which was introduced by Giuseppe Gaeta and Paola Morando in 2004. Moreover, our given method enabled us to solve those constructed variational problems using μ-symmetries.

1. Introduction

The applications of symmetry groups to problems arising in the calculus of variations have their origins in the late papers of Lie, which introduced the subject of “integral invariants” [1]. Lie showed how the symmetry group of a variational problem can be readily computed based on an adaptation of the infinitesimal method used to compute symmetry groups of differential equations [2, Chapter 7].

In some applications, finding equations or variational problems which do not have some desired classical Lie symmetries have remarkable importance. In this paper, we have focused on investigating a method to find such equations or problems. Our given method will be based on 𝜇-symmetries which was introduced by [3] to find Type I hidden symmetries. Hidden symmetries are defined as symmetries that are lost (Type I) or gained (Type II) as the order of an ODE is reduced or as the number of variable of a PDE is reduced [4]. Hidden symmetries are difficult to evaluate since there are no general direct methods for determining them. There are several approaches that we can use to investigate the hidden symmetries.

In 2001, Muriel and Romero introduced 𝜆-symmetries to evaluate Type I hidden symmetries of ODEs [5]. Gaeta and Morando expanded this approach to scalar PDEs and PDEs systems as well as introducing a method for constructing equations without Lie point symmetries in 2004 [3, 6]. These equations have no obvious order reduction (in ODE case) and variable reduction (in scalar PDEs case and PDEs systems) which can be reduced using 𝜇-symmetries. After a few months, in the subsequent paper [7], it was shown that 𝜇-symmetries are essentially equivalent (more precisely: locally, gauge equivalent) to standard symmetries (see especially [7], Theorem 1 and Example 1). The result has been also extended to the variational case in [8, 9].

As before, in this paper we will attempt to present a method for constructing variational problems without having a desired one-parameter transformation group as a variational symmetry. More details, this paper is organized as follows. In Section 2, briefly, the notation of 𝜇-symmetry is introduced in the line of [3, 6]. Section 3 is devoted to applications of symmetry approach in variational problems and some basic concepts of variational calculus. Constructing equations without Lie symmetries and introducing their appropriate 𝜇-symmetries have been focused on in Section 4. Section 5 can be considered as a main part of the paper which is devoted to construct variational problems without desired variational symmetries using results of Section 4 with Theorem 6 and assuming the obtained equations in the preceding section as the Euler-Lagrangian equations of some variational problems with necessary condition (Propositions 9 and 11).

2. 𝜇-Symmetry on Scalar PDEs and PDEs Systems

As before, the starting point will be a discussion of some of the foundational results about 𝜇-symmetry. The reader can consult [3, 6] to gain complete information about these symmetries.

Let 𝜇=𝜆_𝑖𝑑𝑥_𝑖 be a horizontal one form on first order jet space (𝐽¹𝑀,𝜋,𝑀) and compatible with contact structure 𝜀 on 𝐽^𝑘𝑀 for 𝑘≥2. That is, 𝑑𝜇∈𝐽(𝜀), (1) where 𝐽(𝜀) is a Cartan ideal generated by contact structure.

Theorem 1 (see [3]). Condition (1) is equivalent to 𝐷_𝑖𝜆_𝑗−𝐷_𝑗𝜆_𝑖=0, where 𝐷_𝑖 is total derivative with respect to 𝑥_𝑖.

Let 𝑋∶=∑^𝑝_𝑖=1𝜉^𝑖(𝑥,𝑢)𝜕_𝑥𝑖+𝜙(𝑥,𝑢)𝜕_𝑢 be an arbitrary vector field on total space 𝑀. 𝑌∶=𝑋+Ψ_𝑗𝜕𝑢_𝑗 is defined as 𝜇-prolongation of 𝑋, on 𝑘th order jet space 𝐽^𝑘𝑀, if its coefficient satisfies the following 𝜇-prolongation formula: Ψ_𝐽,𝑖=𝐷_𝑖+𝜆_𝑖Ψ_𝐽−𝑢_𝐽,𝑚𝐷_𝑖+𝜆_𝑖𝜉^𝑚. (2)

Remark 2. By setting 𝜇=0 in (2), we can gain ordinary prolongation of 𝑋. So, ordinary prolongation can be assumed as 0-prolongation in 𝜇-prolongation framework.

In the following theorem, the connection between ordinary prolongation and 𝜇-prolongation will be considered.

Theorem 3 (see [3]). Let 𝑋∶=∑^𝑝_𝑖=1𝜉^𝑖𝜕_𝑥𝑖+Φ(𝑥,𝑢)𝜕_𝑢 be a vector field on first order jet space 𝐽¹𝑀; Let 𝑌=𝑋+Ψ_𝑗𝜕_𝑢𝑗 and 𝑋^(𝑘)=𝑋+𝜙_𝑗𝜕_𝑢𝑗 be as 𝜇-prolongation and ordinary prolongation of 𝑋, respectively. Then, we have Ψ_𝐽=Φ_𝐽+𝐹_𝐽, where 𝐹_𝐽 satisfies the recursion relation (with 𝐹₀=0): 𝐹_𝐽,𝑖=(𝐷_𝑖+𝜆_𝑖)𝐹_𝐽+𝜆_𝑖𝐷_𝑖𝑄; where, 𝑄 is a Lie characteristic.

Note, this theorem provides an economic way of computing 𝜇-prolongation of 𝑋, if we already know its ordinary prolongation.

3. Variational Problems and the Lie Standard Reduction Method

In this section, we will point out some aspects of applications of symmetries method to variational problems. The reader can find more information in [2, 10, 11]. The variational problem is finding the extremals (maxima and/or minima) of a functional, 𝔏_Ω(𝑢)=_Ω𝐿𝑥,𝑢^(𝑛)𝑑𝑥,𝛼=1,2,…,𝑞, (3) over some space of functions 𝑢=𝑓(𝑥), 𝑥∈Ω. The integrand 𝐿(𝑥,𝑢^(𝑛)), which is a smooth differential function on the jet space 𝐽^(𝑛), is referred to as the Lagrangian of the variational problem (3). For such problems we can define the Euler-Lagrangian operators as 𝐸_𝛼=_𝐽(−𝐷)_𝐽𝜕_𝑢^𝛼_𝐽,𝛼=1,2,…,𝑞. (4) Applications of the symmetry method on variational problems is motivated by the following theorem of the calculus of variational problems.

Theorem 4 (see [10]). The smooth extremals 𝑢=𝑓(𝑥) of variational problem with the Lagrangian 𝐿(𝑥,𝑢^(𝑛)) must be satisfied in the systems of its related Euler-Lagrange equations: 𝐸_𝛼(𝐿)=Σ_𝐽(−𝐷)_𝐽𝜕𝐿𝜕𝑢^𝛼_𝐽. (5)

The following theorem will be a cornerstone of our approach to the symmetry methods in variational problems.

Theorem 5 (see [2]). 𝐺 is a variational symmetry of variational problem (3) if and only if it is a Lie symmetry of its Euler-Lagrangian equation (4).

4. Euler-Lagrangian Equations without Desired Lie Symmetries

Our given method mainly organized in two steps: first, we characterize equations (scalar PDEs) without desired Lie symmetries (this section); then, we find appropriate variational problems which have them as Euler-Lagrangian equations (next section).

To attain the pointed purpose of this section, we consider 𝑋 as a vector field on first order jet space 𝐽¹𝑀 then determine general scalar PDEs which do not have 𝑋 as Lie symmetries, with admitting it as a 𝜇-symmetry (find more information in [3, 7]).

Consider the scaling vector field 𝑋=𝑥𝜕_𝑥+𝑡𝜕_𝑡+𝑢𝜕_𝑢. We have 𝑄=𝑢−𝑥𝑢_𝑥−𝑡𝑢_𝑡 for the Lie characteristic [2, 7]. The corresponding invariant coordinates (𝑦,𝑣) and the parametric coordinate 𝜎 in 𝑀=(𝑥,𝑡,𝑢) can be chosen as 𝜎=𝑥, 𝑦=𝑡/𝑥 and 𝑣=𝑢/𝑥. The corresponding inverse change of variables is 𝑥=𝜎, 𝑡=𝑦𝜎, 𝑢=𝑣𝜎. Hence, the function 𝑣=𝑣(𝜎,𝑦) is 𝑋-invariant if and only if 𝑣_𝜎=0. The partial derivations of 𝑢 express in the partial derivatives of 𝑣=𝑣(𝜎,𝑦) as 𝑢_𝑥=𝑣+𝜎𝑣_𝜎−𝑦𝑣_𝑦,𝑢_𝑡=𝑣_𝑦. (6) The above can be inverted to give 𝑣_𝑦=𝑢_𝑡,𝑣_𝜎=1𝑥𝑢_𝑥+𝑦𝑢_𝑡−𝑢𝑥. (7) Similar to the above, we have the following expressions for second order derivatives: 𝑢_𝑥𝑥=2𝑣_𝜎+𝜎𝑣_𝜎𝜎−𝑦𝑣_𝜎𝑦+𝑦²𝜎𝑣_𝑦𝑦,𝑢_𝑥𝑡=𝑣_𝜎𝑦−𝑦𝜎𝑣_𝑦𝑦,𝑢_𝑡𝑡=1𝜎𝑣_𝑦𝑦. (8) The central object in this computation is a horizontal one-form 𝜇 on one ordered jet space 𝐽⁽¹⁾𝑀. Here, since the space of independent variables is two dimensional, we have 𝜇=𝜆𝑑𝑥+𝜏𝑑𝑡. Let us come to the second 𝜇-prolongation of 𝑋. For this computation, we can use (2) or the recursion relation in Theorem 3. Hence if we show 𝜇-prolongation of 𝑋 as 𝑌=𝑋+Ψ^𝑥𝜕_𝑢𝑥+Ψ^𝑡𝜕_𝑢𝑡+Ψ^𝑥𝑥𝜕_𝑢𝑥𝑥+Ψ^𝑥𝑡𝜕_𝑢𝑥𝑡+Ψ^𝑡𝑡𝜕_𝑢𝑡𝑡, (9) then, we have Ψ^𝑥=𝜆𝑄,Ψ^𝑡Ψ=𝜏𝑄,_𝑥𝑥=−𝑢_𝑥𝑥+2𝜆𝐷_𝑥𝑄+𝜆²+𝐷_𝑥𝜆Ψ𝑄,_𝑥𝑡=−𝑢_𝑥𝑡+𝜆𝐷_𝑡𝑄+𝜏𝐷_𝑥𝑄+122𝜆𝜏+𝐷_𝑡𝜆+𝐷_𝑥𝜏Ψ𝑄,_𝑡𝑡=−𝑢_𝑡𝑡+2𝜏𝐷_𝑡𝑄+𝜏²+𝐷_𝑡𝜏𝑄. (10) We will now consider the simplest nontrivial choice for 𝜇; that is, 𝜇=𝜆𝑑𝑥, with 𝜆 a real constant. By substituting this 𝜇 in (10), we find Ψ^𝑥=𝜆𝑢−𝑥𝑢_𝑥−𝑡𝑢_𝑡,Ψ^𝑡Ψ=0,_𝑥𝑥=−𝑢_𝑥𝑥−2𝜆𝑥𝑢_𝑥𝑥+𝑡𝑢_𝑥𝑡+𝜆²𝑢−𝑥𝑢_𝑥−𝑡𝑢_𝑡,Ψ^𝑥𝑡=−𝑢_𝑥𝑡−𝜆𝑥𝑢_𝑥𝑡+𝑡𝑢_𝑡𝑡,Ψ^𝑡𝑡=−𝑢_𝑡𝑡. (11) By taking (𝑦,𝑣) as invariants of order zero, (𝜉₁,𝜉₂) of order one, and (𝜂₁,𝜂₂,𝜂₃) of order two, we have 𝑡𝑦=𝑥𝑢,𝑣=𝑥,𝜉₁=𝑢𝑥−𝑡𝑢_𝑡𝑥−𝑢_𝑥𝑒^𝜆𝑥,𝜉₂=𝑢_𝑡,𝜂₁=𝑥𝑢_𝑡𝑡,𝜂₂=𝑥²𝑢_𝑥𝑡+𝑥𝑡𝑢_𝑡𝑡𝑒^𝜆𝑥,𝜂₃=1𝑥(1−𝜆𝑥)𝑢−𝑥𝑢_𝑥+𝜆𝑥𝑡𝑢_𝑡+𝑥²𝑢_𝑥𝑥+𝑥𝑡𝑢_𝑥𝑡+𝑡²𝑢_𝑡𝑡𝑒^2𝜆𝑥. (12)

Theorem 6. Consider the equation Δ∶=𝐹(𝑦,𝑣,𝜉₁,𝜉₂,𝜂₁,𝜂₂,𝜂₃) with 𝐹 as an arbitrary smooth function of its arguments. Let 𝜆 be a real constant. Then, we have the following.(i)Equation Δ admits the vector field 𝑋 as a 𝜇-symmetry with 𝜇=𝜆𝑑𝑥. (ii)For (𝜕𝐹/𝜕𝜉₁)²+(𝜕𝐹/𝜕𝜂₂)²+(𝜕𝐹/𝜕𝜂₃)²≠0, 𝑋 is not an ordinary symmetry of Δ. (iii)The 𝜇-symmetry-reduced equation, providing 𝑋-invariant solutions to Δ, is given by 𝐻𝑦,𝑣,𝜉₂,𝜂₁∶=𝐹𝑦,𝑣,0,𝜉₂,𝜂₁,0,0=0. (13)

Proof. (i) Based on the Lie point symmetry method, a PDE admits 𝑋 as 𝜇-symmetry when we can rewrite it in terms of 𝑋-invariants [2]. Hence, equation Δ admits 𝑋 as 𝜇-symmetry with 𝜇=𝜆𝑑𝑥.
(ii) Using (12), we can conclude that, in solution space, the arguments 𝜉₁, 𝜂₂ and 𝜂₃ depend on 𝜇. So, if 𝐹 depends on this argument, then 𝑋 is not an ordinary symmetry of Δ.
(iii) Finally, note that restriction to solution space does not change the form of 𝜉₂ and 𝜂₁. From the above explicit expression of the invariants, we have that 𝜉₁=𝑒^𝜆𝑥𝑄/𝑥, 𝜂₂=−𝑥𝑒^𝜆𝑥(𝐷_𝑡𝑄), and 𝜂₃=(𝑒^2𝜆𝑥/𝑥)[(1−𝜆𝑥)𝑄−2𝑡(𝐷_𝑡𝑄)−𝑥(𝐷_𝑥𝑄)]. Therefore, on solution space we have 𝜉₁=𝜂₂=𝜂₃=0, and (13) represents indeed the reduced equation for Δ in the space of 𝑋-invariant functions. This proves point (iii) and hence completes the proof.

We would like to note that, as claimed by (13), the reduced equation corresponds to the one which would be obtained by the standard symmetry reduction (under 𝑋) of the equation 𝐹[𝑦,𝑣,𝜉̂₁,𝜉₂,𝜂₁,𝜂₂,𝜂₃]=0 to solution space, where 𝜉̂₁ is obtained from 𝜉₁ by setting 𝜆=0, and the like for 𝜂₂ and 𝜂₃.

Finally, let us discuss a completely concrete example; albeit, the procedure here is completely analogous to the one to be followed for standard symmetries. Consider equation Δ∶=𝜂₁−𝜂₂=0. This is written as 𝑥𝑢_𝑡𝑡−𝑥²𝑢_𝑥𝑡+𝑥𝑡𝑢_𝑡𝑡𝑒^𝜆𝑥=0, (14) in the original coordinates; in the adapted ones it reads 𝑣_𝑦𝑦−𝜎²𝑣_𝜎𝑦𝑒^𝜎𝜆=0. (15) The corresponding reduced equation is 𝑣_𝑦𝑦=0, (16) and the general solution to this is 𝑣(𝑦)=𝑐₁+𝑐₂𝑦, where 𝑐_𝑖 are real constants. Going back to the original coordinates, the corresponding solutions are 𝑢(𝑥,𝑡)=𝑐₁𝑥+𝑐₂𝑡.

Remark 7. Note, unlike the Lie method, we focus on solutions which have not admitted the desired Lie symmetries. For instance, here, beside finding the solution 𝑢(𝑥,𝑡)=𝑐₁𝑥+𝑐₂𝑡 for (14), we found that for some 𝑠, this does not remain again a solution under scaling transformation (𝑥,𝑡,𝑢)↦(𝑒^𝑠𝑥,𝑒^𝑠𝑡,𝑒^𝑠𝑢).

5. 𝜇-Symmetry on Variational Problems

In the calculus of variations, the problem of characterizing systems of differential equations which are the Euler-Lagrange equations for some variational problems is known as the inverse problem [7, 8]. There are different approaches to perform the inverse problem [8, 9]. Here, in order to keep the scope manageable, we use the direct method.

In this section, applying the preceding results, we construct an example of a variational problem which its Euler-Lagrange equation has no desired Lie symmetries. For this purpose, first, we assume that Δ is an equation without desired Lie symmetry (here, scaling symmetry); then, we find appropriate variational problems which have equation Δ as a Euler-Lagrangian equation. Also, in Proposition 11, we will introduce the class of variational problems without having any nontrivial Lie variational symmetries.

Example 8. Consider this equation 𝑥𝑢_𝑡𝑡−𝑒^𝑥𝑥²𝑢_𝑥𝑡+𝑥𝑡𝑢_𝑡𝑡=0, (17) it can be written as: 𝜂₁−𝜂₂=0, using given expressions in (12) with 𝜆=1. Based on Theorem 6 this equation has scaling symmetry: (𝑥,𝑡,𝑢)↦(𝑒^𝑠𝑥,𝑒^𝑠𝑡,𝑒^𝑠𝑢) as 𝜇-symmetry with 𝜇=𝑑𝑥, and this transformation is not its Lie symmetry. Let it be an Euler-Lagrange equation for some second-order variational problem; so, from (4) we have 𝜕𝐿𝜕𝑢−𝐷_𝑥𝜕𝐿𝜕𝑢_𝑥−𝐷_𝑡𝜕𝐿𝜕𝑢_𝑡+𝐷²_𝑥𝜕𝐿𝜕𝑢_𝑥𝑥+𝐷_𝑥𝐷_𝑡𝜕𝐿𝜕𝑢_𝑥𝑡+𝐷²_𝑡𝜕𝐿𝜕𝑢_𝑡𝑡=𝑥𝑡𝑢_𝑡𝑡−𝑒^𝑥𝑥²𝑢_𝑥𝑡+𝑥𝑡𝑢_𝑡𝑡. (18) By solving this equation, we have 𝐿₁𝑥,𝑡,𝑢,𝑢_𝑥,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑡,𝑢_𝑡𝑡=12𝑢𝑥𝑡𝑢_𝑡𝑡−𝑒^𝑥𝑥²𝑢_𝑥𝑡+𝑥𝑡𝑢_𝑡𝑡. (19)

As a result of the above and using Theorem 5 and Corollary 7.4 in [2], we can find the following.

Proposition 9. Any variational problem with the Lagrangian 𝐿=𝐿₁+Div𝜉 for arbitrary smooth function 𝜉 does not admit scaling transformation (𝑒^𝑠𝑥,𝑒^𝑠𝑡,𝑒^𝑠𝑢) as a variational symmetry. Moreover, this transformation is a 𝜇-symmetry of its Euler-Lagrangian equation with 𝜇=𝑑𝑥.

Example 10. Consider the equation 𝑢_𝑥𝑥+𝑥²4𝑢³1+𝑢+2𝑢=0. (20) This equation appears in Theorem 4.3 of [5] as an equation which can be integrated using the 𝜆-symmetry method with 𝜆=𝑥/𝑢² and 𝑋=𝑢𝜕_𝑢 but lacks nontrivial symmetries. Let it be an Euler-Lagrange equation for some second-order variational problem; so, from (4) we have 𝜕𝐿𝜕𝑢−𝐷_𝑥𝜕𝐿𝜕𝑢_𝑥=𝑢_𝑥𝑥+𝑥²4𝑢³1+𝑢+,2𝑢 (21) by solving (21), we find the following Lagrangian: 𝐿₂𝑥,𝑢,𝑢_𝑥=12𝑢²𝑢_𝑥+𝑢²+ln𝑢−𝑢²_𝑥−𝑥²4𝑢². (22)

Using (Theorem 5) and Corollary 7.4 of [2], the following will hold.

Proposition 11. Any variational problem with the Lagrangian 𝐿=𝐿₂+Div𝜉 for arbitrary smooth function 𝜉 has no nontrivial variational Lie symmetries, where its Euler-Lagrangian equation has 𝑋=𝑢𝜕_𝑢 as a 𝜆-symmetry with 𝜆=𝑥/𝑢².

6. Conclusions

In this paper, first, we construct some Euler-Lagrange equations with no desired symmetries as well as introducing appropriate 𝜇-symmetries to reducing them; next, we find their related variational problems without having desired variational symmetries. Indeed, in Propositions 9 and 11, we introduced two class of variational problems which have no scaling variational symmetry and nontrivial Lie symmetries, respectively.

Acknowledgment

The authors would like to thank the referees for constructive suggestions and helpful discussions which have materially improved the presentation of the paper.